QUANTITATIVE STABILITY VIA THE METHOD OF MOVING PLANES: APPROXIMATE SYMMETRY FOR OVERDETERMINED AND RIGIDITY PROBLEMS

Since its introduction thanks to the work of Alexandrov, the method of moving planes has seen a widespread use in various applications of geometric analysis. Brought to the attention of the PDE community with the seminal work of Serrin, it has been a useful tool to prove a large variety of results, including symmetry results for overdetermined and rigidity problems. This thesis investigates three such problems from a quantitative viewpoint, by employing the method of moving planes and developing tools and techniques to prove symmetry and approximate symmetry results.